Multiscale anaylses of permeability in porous and fractured media

Abstract

It has been shown by Neuman [1990], Di Federico and Neuman [1997, 1998a,b] and Di Federico et al. [1999] that observed multiscale behaviors of subsurface fluid flow and transport variables can be explained within the context of a unified stochastic framework, which views hydraulic conductivity as a random fractal characterized by a power variogram. Any such random fractal field is statistically nonhomogeneous but possesses homogeneous spatial increments. When the field is statistically isotropic, it is associated with a power variogram γ(s) = Cs²ᴴ where C is a constant, s is separation distance, and If is a Hurst coefficient (0 < H< 1). If the field is Gaussian it constitutes fractional Brownian motion (fBm). The authors have shown that the power variogram of a statistically isotropic or anisotropic fractal field can be constructed as a weighted integral from zero to infinity of exponential or Gaussian vario grams of overlapping, homogeneous random fields (modes) having mutually uncorrelated increments and variance proportional to a power 2H of the integral (spatial correlation) scale. Low- and high-frequency cutoffs are related to length scales of the sampling window (domain) and data support (sample volume), respectively. Intermediate cutoffs account for lacunarity due to gaps in the multiscale hierarchy, created by a hiatus of modes associated with discrete ranges of scales. In this dissertation, I investigate the effects of domain and support scales on the multiscale properties of random fractal fields characterized by a power variogram using real and synthetic data. Neuman [1994] and Di Federico and Neuman [1997] have concluded empirically, on the basis of hydraulic conductivity data from many sites, that a finite window of length-scale L filters out (truncates) all modes having integral scales λ larger than λ = μL where μ ≃ 1/3. I confii in their finding computationally by generating truncated fBm realizations on a large grid, using various initial values of μ, and demonstrating that μ ≃ 1/3 for windows smaller than the original grid. My synthetic experiments also show that generating an fl3m realization on a finite grid using a truncated power variogram yields sample variograms that are more consistent with theory than those obtained when the realization is generated using a power variogram. Interpreting sample data from such a realization using wavelet analysis yields more reliable estimates of the Hurst coefficient than those obtained when one employs variogram analysis. Di Federico et al. [1997] developed expressions for the equivalent hydraulic conductivity of a box-shaped support volume, embedded in a log-hydraulic conductivity field characterized by a power variogram, under the action of a mean uniform hydraulic gradient. I demonstrate that their expression and empirically derived value of μ ≃ 1/3 are consistent with a pronounced permeability scale effect observed in unsaturated fractured tuff at the Apache Leap Research Site (ALRS) near Superior, Arizona. I then investigate the compatibility of single-hole air permeability data, obtained at the ALRS on a nominal support scale of about 1 m, with various scaling models including fBm, fGn (fractional Gaussian noise), fLm (fractional Lévy motion), bfLm (bounded fractional Lévy motion) and UM (Universal Multifractals). I find that the data have a Lévy-like distribution at small lags but become Gaussian as the lag increases (corresponding to bfLm). Though this implies multiple scaling, it is not consistent with the UM model, which considers a unique distribution. If one nevertheless applies a UM model to the data, one obtains a very small codimension which suggests that multiple scaling is of minor consequence (applying the UM model to permeability rather than log-permeability data yields a larger codimension but is otherwise not consistent with these data). Variogram and resealed range analyses of the log-permeability data yield comparable estimates of the Hurst coefficient. Resealed range analysis shows that the data are not compatible with an fGn model. I conclude that the data are represented most closely by a truncated fBm model.